Abstract
In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.
Highlights
We discretize (1.1) on a time-dependent spatial grid and add an extra equation that aims to distribute the gridpoints in such a way that the arclength of the solution is equal between any two consecutive gridpoints
Our main contribution in this series of papers is to show that the resulting coupled semidiscrete system is well-posed and admits solutions that can be interpreted as travelling waves
Fitzhugh [22,23] was able to effectively describe the propagation of signal spikes through nerve fibres. Sparked by his interest in morphogenesis, Turing [48] described the famous bifurcation through which equilibria of general two-component reaction–diffusion systems can destabilize and generate spatially periodic structures such as spots and stripes. These early results led to the development of many important technical tools that today are indispensable to the field of dynamical systems
Summary
We discretize (1.1) on a time-dependent spatial grid and add an extra equation that aims to distribute the gridpoints in such a way that the arclength of the solution is equal between any two consecutive gridpoints. Fitzhugh [22,23] was able to effectively describe the propagation of signal spikes through nerve fibres Sparked by his interest in morphogenesis, Turing [48] described the famous bifurcation through which equilibria of general two-component reaction–diffusion systems can destabilize and generate spatially periodic structures such as spots and stripes. These early results led to the development of many important technical tools that today are indispensable to the field of dynamical systems. The development of Evans function [34] and semigroup theory [45] was heavily influenced by the desire to analyze the stability of many of these localized structures
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